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## Curriculum and Assessment in an Age of Computer Algebra 
Systems. ERIC Digest.

The use of Computer Algebra System (CAS) technology in the teaching of 
mathematics is entering a new and crucial stage. A considerable amount of work 
to date has focused on support of existing curriculum topics and existing 
curriculum models. Kutzler's seminal book on teaching and learning using DERIVE 
(1996) includes the phrase "here is an example of how CAS can be used to support 
an existing curriculum topic". Arguments for the use of CAS as an effective tool 
in supporting teaching and learning of mathematics are well established. Now 
comes the real work.

Curriculum models and assessment of curriculums are the structures that 
schools and examining boards create to educate students. The two are deeply 
connected and the real work of CAS advocates is now to develop curriculum models 
in which the use of CAS is an integral part, not just of the practice of the 
curriculum, but, of the conception of the curriculum. A thorough examination of 
topics must be carried out to decide what really has to stay for conceptual 
reasons and what is still extant only because of tradition.

A crucial part of this process is a demonstration of assessment using CAS. 
Unless instruments can be found to measure the achievement, or lack of 
achievement, of that learning which is possible through the use of CAS it is 
destined to remain as a prop occasionally brought out to support and bolster 
existing methods of teaching and learning.

We should take the current topics, the current models and using those topics 
see how we can begin to assess students who are answering examination questions 
with CAS as an aid. This exercise serves the very useful purpose of bringing to 
light the relative triviality, in a CAS age, of some common current questions 
and forces us to examine their real significance as mathematical concepts. The 
close scrutiny of current examination questions also affords the possibility for 
the incorporation of more complicated questions previously beyond the scope of a 
time-limited written examination.

### RESEARCH ON ASSESSMENT WITH CAS TECHNOLOGY

Several have examined the impact of CAS on current national examinations with 
questions posed about the future of such examinations in the CAS age. Aogain 
(2000) looked at the Irish Leaving Certificate Examinations for 1999 and 
classified questions as trivial, easy, difficult or CAS proof. He proposes a 
calculation of an "index of suitability" for examinations that could be applied 
to any examination.

A review of examinations in several European countries was undertaken by 
Drijvers (1998) who noticed several approaches and classified them according to 
the extent to which technology is allowed and, if allowed, the reward available 
to those who use it.

Brown (1999) compared some examination questions from Australia, Denmark and 
the USA and noted the clear difficulties that examining boards are having in 
finding an appropriate approach to assessment. He offers the view that "There 
will undoubtedly be an evolution in the development of setting questions that 
encourage the use of technology while ensuring that the primary focus of the 
mathematics assessment will remain on assessing mathematical concepts and 
understanding." Others working in this area include Fazio (2000), Lokar & 
Lokar (2000) & Ruthven (1997).

Kutzler (2000) makes the case for two-tier examinations, one part with no 
technology allowed and a second part with any and every technology allowed. As 
Kutzler says "Two-tiered exams would be a well-balanced compromise meeting both 
the desires of technology supporters and the reservations of those who are 
concerned about the use of technology in the classroom."

### TEACHER ACCOUNTS OF ASSESSMENT WITH CAS TECHNOLOGY

Torres-Skoumal (2000) found group assessment very effective both for 
improving technical competencies and mathematics achievement. Other benefits she 
observed were social integration, student awareness of assessment procedures, 
and improvement in use of mathematical language.

Kempski (1988) uses DERIVE in the teaching and assessment of mathematics. 
Based on several years of experience he suggested that "students who best use 
the software are those who are more able and confident in Mathematics." Kempski 
also considered the way that students used DERIVE in examinations and was 
interested to observe that "more often than not, they demonstrated an inability 
to make the best use of the software and did not even take full advantage of the 
routine menu commands."

The proceedings of the 6th ACDCA Summer Academy in Portoroz, Slovenia, 
published under the title "Exam Questions & Basic Skills in 
Technology-Supported Mathematics Teaching (Proceedings Portoroz 2000)" 
(Kokol-Voljc et al, 2000) contains many more articles related to teaching, 
learning and assessment in the CAS age.

### CURRICULUM ISSUES

Heugl (2000) has listed what he sees as seven important competencies in 
algebra, such as the competence of recognizing structures, recognizing 
equivalence of terms, and visualization. He discusses the seven competencies in 
terms of the impact of CAS on them, and he suggests several different 
examination models which may prove more responsive to the impact of CAS than 
traditional examinations.

Finally, a very controversial addition to the debate is the paper arising 
from the discussion between Herget, Heugl, Kutzler and Lehmann (2000) about what 
they consider to be the essential skills in arithmetic and algebra in the CAS 
age. They imagine an environment of a technology-free examination and classify 
problems into essentially two types: those questions which a student would be 
expected to answer in such a technology-free examination (i.e. without any 
calculator or computer), and those questions which would not be asked in such an 
examination. A third group in the classification are those questions about which 
the authors have doubts as to where to classify them. The authors stress that 
those questions that they would not ask in a technology-free examination they 
find inappropriate for any examination, even one where powerful technology is 
allowed. As they say "we would not ask these questions in a technology-supported 
exam either, because these questions appear useless as such, their best use 
might be to test how well a student can operate a calculator." The paper is 
intentionally controversial and as the authors themselves say "we deliberately 
wanted to be provocative and shake the mainstay of traditional mathematics 
teaching."

### REFERENCES

Cabezas, J. & Roanes-Lozano, E. (1998). Do Computer Algebra Systems 
Change the Order in Which we Should Teach Mathematics? Proceedings of the 3rd 
International Derive and TI-92 Conference. Gettysburg, PA.

Drijvers, P. (1998). Assessment and the new Technologies. The international 
"Journal of Computer Algebra in Mathematics Education", 5(2), 81-93.

Fazio, R. (2000). The Mathematics Final Exam Questions for the Italian 
Experimental Scientific Liceo Discussed and Solved with Derive 5. In 
Kokol-Voljc, V., Kutzler, B., Lokar, M. & Palcic, J. (2000).

Heid, M K. (1988). Resequencing Skills and Concepts in Applied Calculus using 
the Computer as a Tool. "Journal of Research in Mathematics Education," 19(1), 
3-21.

Herget, W., Heugl, H. Kutzler, B., & Lehmann, E. (2000). Indispensable 
Manual Calculation Skills in a CAS Environment. In Kokol-Voljc, V., Kutzler, B., 
Lokar, M. & Palcic, J. (2000). Also available at 
http://www.kutzler.com/article/art_indi/indisp.htm

Heugl, H. (2000). New emphasis of Fundamental Algebraic Competence and its 
Influence in Exam Situation. In Kokol-Voljc, V., Kutzler, B., Lokar, M. & 
Palcic, J. (2000).

Kempski, B. (1998). The Use of DERIVE in Assessed Student Work. "Proceedings 
of the 3rd International Derive and TI-92 Conference." Gettysburg, PA.

Kokol-Voljc, V. (1999a). Exam Questions when using CAS for School Mathematics 
Teaching. "International Journal of Computer Algebra in Mathematics Education." 
7(1), 63-76.

Kokol-Voljc, V. (1999b). "Exam Questions for DERIVE &TI-92/89." bk 
teachware, Hagenberg (in German). ISBN 3-901769-23-4.

Kokol-Voljc, V., Kutzler, B., Lokar, M. & Palcic, J. (2000). "Exam 
Questions and Basic Skills in Technology-Supported Mathematics Teaching" 
(Proceedings, July 2-5, 2000, Portoroz, Slovenia). bk-teachware, Hagenberg. ISBN 
3-901769-33-1.

Kutzler, B. (1996). "Improving Mathematics Teaching with DERIVE." 
Chartwell-Bratt, Bromley, UK. ISBN 0-86238-422-2.

Kutzler, B. (2000). Two-Tier Examinations as a Way to Let Technology. In 
Kokol-Voljc, V., Kutzler, B., Lokar, M. & Palcic, J. (2000). Also available 
at http://www.kutzler.com/article/art_exam/exams.htm

Lokar, M. & Lokar, M. (2000). Slovene Final External Examination -- 
Matura in View of Computer Algebra Systems. In Kokol-Voljc, V., Kutzler, B., 
Lokar, M. & Palcic, J. (2000).

Mac Aogain, E. (2000). Assessment in the CAS age: an Irish perspective. In 
Kokol-Voljc, V., Kutzler, B., Lokar, M. & Palcic, J. (2000).

McCrae, B. & Stacey, K. (2000). "Computer Algebra Systems in Schools -- 
Curriculum, Assessment and Teaching Project."

Ruthven, K. (1997). "Computer Algebra Systems in Advanced-Level Mathematics A 
report to SCAA. Cambridge: University of Cambridge." Torres-Skoumal, M. (2000). 
Alternative Assessment Model. In Kokol-Voljc, V., Kutzler, B., Lokar, M. & 
Palcic, J. (2000).

Michael Meagher is a Doctoral student in Mathematics Education at Ohio State 
University and can be contacted at [email protected].

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This digest was funded by the Office of Educational Research and Improvement, 
U.S. Department of Education, under contract no. ED-99-CO-0024. Opinions 
expressed in this digest do not necessarily reflect the positions or policies of 
OERI or the U.S. Department of Education.

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